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| 183_notes:mp_multi [2015/09/21 02:12] – [Multi-particle Systems] caballero | 183_notes:mp_multi [2021/04/01 01:50] (current) – [The Momentum Principle for Multiple Particles] stumptyl |
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| | Section 3.10 and 3.11 in Matter and Interactions (4th edition) |
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| ===== The Momentum Principle in Multi-Particle Systems ===== | ===== The Momentum Principle in Multi-Particle Systems ===== |
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| Until now, you've only considered systems of a [[183_notes:momentum_principle#system_and_surroundings|single particle]]. This greatly simplifies the concept of a system, but doesn't really communicate why the concept of a system is so essential to physics. When you have several objects in a system, we refer to these as "multi-particle systems." In these notes, you will read about multi-particle systems, why it is often advantageous to make use of them, and how the [[183_notes:momentum_principle|momentum principle]] is defined for systems with several objects. | Until now, you've only considered systems of a [[183_notes:momentum_principle#system_and_surroundings|single particle]]. This greatly simplifies the concept of a system, but doesn't really communicate why the concept of a system is so essential to physics. When you have several objects in a system, we refer to these as "multi-particle systems." **In these notes, you will read about multi-particle systems, why it is often advantageous to make use of them, and how the [[183_notes:momentum_principle|momentum principle]] is defined for systems with several objects.** |
| ==== Lecture Video ==== | ==== Lecture Video ==== |
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| {{youtube>GIdy_KaFqVs?large}} | {{youtube>GIdy_KaFqVs?large}} |
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| ==== Multi-particle Systems ==== | ===== Multi-particle Systems ===== |
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| Earlier you read about the [[183_notes:momentum_principle#system_and_surroundings|concept of a system]] as defining what objects you want to predict or explain the motion of. Anything outside of your system is the surroundings and can influence the system by changing some of its properties ([[183_notes:momentum|momentum]], [[183_notes:point_particle|energy]], and [[183_notes:ang_momentum|angular momentum]]). Remember that the choice of system is arbitrary to the extent that you only care about predicting or explaining the motion of objects in your system. | Earlier you read about the [[183_notes:momentum_principle#system_and_surroundings|concept of a system]] as defining what objects you want to predict or explain the motion of. Anything outside of your system is the surroundings and can influence the system by changing some of its properties ([[183_notes:momentum|momentum]], [[183_notes:point_particle|energy]], and [[183_notes:ang_momentum|angular momentum]]). Remember that the choice of system is arbitrary to the extent that you only care about predicting or explaining the motion of objects in your system. |
| Sometimes, it is advantageous (or necessary) to include more than one object in your system. Doing so, can often simplify things (e.g., when the momentum of the system does not change). To be clear, this is not just a [[https://en.wikipedia.org/wiki/Sleight_of_hand|sleight of hand]], but really about what motion you care about predicting or explaining. | Sometimes, it is advantageous (or necessary) to include more than one object in your system. Doing so, can often simplify things (e.g., when the momentum of the system does not change). To be clear, this is not just a [[https://en.wikipedia.org/wiki/Sleight_of_hand|sleight of hand]], but really about what motion you care about predicting or explaining. |
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| === Linear Momentum of a Multi-particle System === | ==== Linear Momentum of a Multi-particle System ==== |
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| [{{ 183_notes:conservation_of_momentum1.png?300|A system of 3 particles, each with its own mass ($m_i$) and velocity ($\vec v_i$).}}] | [{{ 183_notes:conservation_of_momentum1.png?300|A system of 3 particles, each with its own mass ($m_i$) and velocity ($\vec v_i$).}}] |
| $$\dfrac{\Delta \vec{p}}{\Delta t} = \vec{F}_{net}$$ | $$\dfrac{\Delta \vec{p}}{\Delta t} = \vec{F}_{net}$$ |
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| As you have read, the rate of change of the momentum for a single particle is due to the interactions that the object has with its surroundings -- these interactions add to give rise to a net //external// force. The word external is key because the interactions must be outside the system of the single object. //An object cannot exert forces on itself in ways to change its own momentum.// | As you have read, the rate of change of the momentum for a single particle is due to the interactions that the object has with its surroundings -- these interactions add to give rise to a net //external// force. The word external is key because the interactions must be outside the system of the single object. //**An object cannot exert forces on itself in ways to change its own momentum.**// |
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| In a multi-particle system, objects within the system interact with each other and exert forces on each other. However, the total momentum of the system can only change due to //external// forces. The momentum principle for a multi-particle system states that the change in the system's momentum ($\Delta \vec{p}_{sys}$) arises from interactions with the system's surroundings ($\vec{F}_{surr}\Delta t$): | In a multi-particle system, objects within the system interact with each other and exert forces on each other. However, the total momentum of the system can only change due to __external forces__. The momentum principle for a multi-particle system states that the change in the system's momentum ($\Delta \vec{p}_{sys}$) arises from interactions with the system's surroundings ($\vec{F}_{surr}\Delta t$): |
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| $$\Delta \vec{p}_{sys} = \vec{F}_{surr}\Delta t$$ | $$\Delta \vec{p}_{sys} = \vec{F}_{surr}\Delta t$$ |